US7856129B2 - Acceleration of Joseph's method for full 3D reconstruction of nuclear medical images from projection data - Google Patents
Acceleration of Joseph's method for full 3D reconstruction of nuclear medical images from projection data Download PDFInfo
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- US7856129B2 US7856129B2 US11/716,359 US71635907A US7856129B2 US 7856129 B2 US7856129 B2 US 7856129B2 US 71635907 A US71635907 A US 71635907A US 7856129 B2 US7856129 B2 US 7856129B2
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- 238000000034 method Methods 0.000 title claims abstract description 67
- 230000001133 acceleration Effects 0.000 title 1
- 230000004044 response Effects 0.000 claims abstract description 12
- 238000010008 shearing Methods 0.000 claims abstract description 8
- 238000002059 diagnostic imaging Methods 0.000 claims description 14
- 238000002603 single-photon emission computed tomography Methods 0.000 claims description 6
- 238000002600 positron emission tomography Methods 0.000 description 11
- 230000006870 function Effects 0.000 description 7
- 230000008569 process Effects 0.000 description 7
- 238000009499 grossing Methods 0.000 description 5
- 239000012217 radiopharmaceutical Substances 0.000 description 5
- 229940121896 radiopharmaceutical Drugs 0.000 description 5
- 230000002799 radiopharmaceutical effect Effects 0.000 description 5
- 230000000694 effects Effects 0.000 description 4
- 238000003384 imaging method Methods 0.000 description 4
- 238000011282 treatment Methods 0.000 description 3
- 230000007423 decrease Effects 0.000 description 2
- 230000001419 dependent effect Effects 0.000 description 2
- 238000003745 diagnosis Methods 0.000 description 2
- 239000003814 drug Substances 0.000 description 2
- 230000005251 gamma ray Effects 0.000 description 2
- 239000000700 radioactive tracer Substances 0.000 description 2
- 238000007476 Maximum Likelihood Methods 0.000 description 1
- 238000013527 convolutional neural network Methods 0.000 description 1
- 230000006378 damage Effects 0.000 description 1
- 238000001514 detection method Methods 0.000 description 1
- 201000010099 disease Diseases 0.000 description 1
- 208000037265 diseases, disorders, signs and symptoms Diseases 0.000 description 1
- 229940079593 drug Drugs 0.000 description 1
- 238000001914 filtration Methods 0.000 description 1
- 230000003993 interaction Effects 0.000 description 1
- 230000002028 premature Effects 0.000 description 1
- 230000005258 radioactive decay Effects 0.000 description 1
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- G—PHYSICS
- G06—COMPUTING; CALCULATING OR COUNTING
- G06T—IMAGE DATA PROCESSING OR GENERATION, IN GENERAL
- G06T11/00—2D [Two Dimensional] image generation
- G06T11/003—Reconstruction from projections, e.g. tomography
- G06T11/005—Specific pre-processing for tomographic reconstruction, e.g. calibration, source positioning, rebinning, scatter correction, retrospective gating
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- A—HUMAN NECESSITIES
- A61—MEDICAL OR VETERINARY SCIENCE; HYGIENE
- A61B—DIAGNOSIS; SURGERY; IDENTIFICATION
- A61B6/00—Apparatus or devices for radiation diagnosis; Apparatus or devices for radiation diagnosis combined with radiation therapy equipment
- A61B6/02—Arrangements for diagnosis sequentially in different planes; Stereoscopic radiation diagnosis
- A61B6/03—Computed tomography [CT]
- A61B6/037—Emission tomography
Definitions
- the current invention is in the field of medical imaging, and in particular pertains to reconstruction of tomographic images from acquired projection data obtained by an imaging apparatus.
- Medical imaging is one of the most useful diagnostic tools available in modern medicine. Medical imaging allows medical personnel to non-intrusively look into a living body in order to detect and assess many types of injuries, diseases, conditions, etc. Medical imaging allows doctors and technicians to more easily and correctly make a diagnosis, decide on a treatment, prescribe medication, perform surgery or other treatments, etc.
- NM imaging nuclear medical
- PET positron emission tomography
- SPECT single photon emission computed tomography
- a PET camera works by detecting pairs of gamma ray photons in time coincidence.
- the two photons arise from the annihilation of a positron and electron in the patient's body.
- the positrons are emitted from a radioactive isotope that has been used to label a biologically important molecule (a radiopharmaceutical).
- a radiopharmaceutical a radiopharmaceutical
- Hundreds of millions such decays occur per second in a typical clinical scan.
- the rate of detection of such coincident pairs is proportional to the amount of emission activity, and hence the molecule, along the line connecting the two detectors at the respective points of gamma ray interaction.
- the detectors are typically arranged in rings around the patient.
- a set of projection views can be formed, each element of which represents a line integral, or sum, of the emission activity in the patient's body along a well defined path.
- These projections are typically organized into a data structure called a sinogram, which contains a set of plane parallel projections at uniform angular intervals around the patient. A three dimensional image of the radiopharmaceutical's distribution in the body then can be reconstructed from these data.
- a SPECT camera functions similarly to a PET camera, but detects only single photons rather than coincident pairs. For this reason, a SPECT camera must use a lead collimator with holes, placed in front of its detector panel, to pre-define the lines of response in its projection views.
- One or more such detector panel/collimator combinations rotates around a patient, creating a series of planar projections each element of which represents a sum of the emission activity, and hence biological tracer, along the line of response defined by the collimation.
- these data can be organized into sinograms and reconstructed to form an image of the radiopharmaceutical tracer distribution in the body.
- a conventional reconstruction step involves a process known as back-projection.
- back-projection an individual data sample is back-projected by setting all the image pixels along the line of response pointing to the sample to the same value.
- a back-projection is formed by smearing each view back through the image in the direction it was originally acquired. The back-projected image is then taken as the sum of all the back-projected views. Regions where back-projection lines from different angles intersect represent areas which contain a higher concentration of radiopharmaceutical.
- Filtered back-projection is a technique to correct the blurring encountered in simple back-projection.
- Each projection view is filtered before the back-projection step to counteract the blurring point spread function. That is, each of the one-dimensional views is convolved with a one-dimensional filter kernel (e.g. a “ramp” filter) to create a set of filtered views. These filtered views are then back-projected to provide the reconstructed image, a close approximation to the “correct” image.
- a one-dimensional filter kernel e.g. a “ramp” filter
- the basic assumption is that the object considered truly consists of an array of N ⁇ N square pixels, with the image function ⁇ (x, y) assumed to be constant over the domain of each pixel.
- the method proceeds by evaluating the length of intersection of each ray with each pixel, and multiplying the value of the pixel (S).
- a second class of algorithms for calculating projections is the forward projection method. This method is literally the adjoint of the process of “back projection” of the FBP reconstruction algorithm. The major criticism of this algorithm is that the spatial resolution of the reprojection is lessened by the finite spacing between rays. Furthermore, increasing the number of pixels does not contribute to a reduction in this spacing, but does greatly increase processing time.
- a method for reconstructing a tomographic image from projection data by interpolating an oblique ray or line of response (LOR) through a rectangular volume having the steps of: interpolating all the direct rays in a rectangular volume, creating a projected ray by projecting the oblique ray onto a surface of the rectangular volume, matching the projected ray to a coinciding interpolated direct ray, shearing the rectangular volume to match the projected ray, and interpolating the oblique ray in the sheared volume.
- a method for interpolating a number of oblique rays through a rectangular volume having the steps of: interpolating all the direct rays in a rectangular volume, creating a plurality of projected rays for each oblique ray by projecting the oblique rays onto a surface of the rectangular volume, matching each projected ray to a coinciding interpolated direct ray, creating a plurality of sheared volumes by shearing the rectangular volume to match the projected rays, and interpolating each oblique ray in each sheared volume.
- a method for interpolating at least two oblique rays of opposite polar angle through a rectangular volume having the steps of: interpolating all the direct rays in a rectangular volume, projecting the of oblique rays of opposite polar angle onto a surface of the rectangular volume, matching the projected rays to a coinciding interpolated direct ray, creating sheared volumes for each projected ray by shearing the rectangular volume to match the projected rays, interpolating one oblique ray of opposite polar angle in its corresponding sheared volume, and applying the interpolated value to the rest of the oblique rays of opposite polar angle.
- the system includes a medical imaging device, a processor, and software running on the processor that executes the methods of the present invention.
- FIG. 1 is a representation of Joseph's Method for two dimensional interpolation.
- FIGS. 2A-C are three dimensional, front, and side views, respectively, of a oblique ray in a rectangular volume.
- FIG. 3 is a representation of Joseph's Method for three dimensional interpolation.
- FIG. 4A-B are front and side views, respectively, of the three dimensional interpolation of FIG. 3 .
- FIG. 5 is a front view of a sheared space for the three dimensional interpolation of FIG. 3 in accordance with the present invention.
- FIG. 6 is a three dimensional space with two opposite polar angle rays passing through it.
- FIGS. 7A and 7B are front and side views, respectively, of the three dimensional space of FIG. 6 .
- FIGS. 8A and 8B are the front views of the sheared space for the two rays in FIG. 6 .
- FIG. 9 is a flow chart of a method according to the present invention.
- FIG. 10 is a system using the methods of the present invention.
- FIGS. 11A and 11B are top and cross-sectional views, respectively, of a cylindrical PET scanner with multiple detector rings, which is applicable to the present invention.
- Joseph's Method is a method for reprojecting rays through pixel images using line integrals. The basic assumption is that the image is a smooth function of x and y sampled on a grid of points in (x,y) space.
- the line integral desired is related to an integral over either x or y depending on whether ray 120 's direction lies closer to the x or y axis, that is
- the one dimensional integral is approximated by a simple sum, such as a Riemann sum; for example, the x-direction integral becomes
- Interpolation enters in two senses: 1) explicitly, in the use of fraction ⁇ n to estimate the value of ⁇ ( x n ,y ( x n )) ⁇ (1 ⁇ n ) P n,n′ + ⁇ n P n,n′+1 and 2) implicitly in the sense that the summation above is the application of the trapezoidal rule to numerically estimate the one dimensional (x) integral.
- the treatment of the endpoints T 1 and T N depend on the application. In some situations, they may be taken to be zero if outside the object images. For applications to heart-isolating algorithms, it is necessary to make them proportional to the length of intersection of the ray with the first and last pixels.
- each ray 120 receives information from the two nearest pixels 130 A and 130 B.
- the distances 160 A and 160 B between the centers of pixels 130 A and 130 B and the point 150 where ray 120 intersects the horizontal line 140 passing through the center of pixels 130 A and 130 B define the interpolation coefficients.
- FIGS. 11A-11B is a schematic representation of a cylindrical PET scanner 1101 , and its cross-section, respectively.
- the PET scanner 1101 includes multiple detector rings, such as rings 1102 - 1105 .
- Oblique rays 1106 and 1107 correspond to various non-zero ring difference. For example, ray 1106 extends between rings 1104 and 1105 , while ray 1107 extends between rings 1103 and 1104 .
- Rays 1106 and 1107 have the same transaxial coordinates (in the x-y plane) as direct rays 1108 , which extends within the same detector ring 1102 . There is also an axial translation symmetry for all rays with the same ring difference.
- FIG. 2A is an example of an oblique segment ray 220 in three dimensional space 210 .
- Oblique segment ray 220 receives information from the four nearest voxels (i.e., volume elements or three dimensional pixels) 215 A, 215 B, 215 C and 215 D in an (x,y,z) image volume: the four voxels can be broken down into four pixels, two pixels 230 A and 230 B in the x direction ( FIG. 2B ), and two pixels 231 A and 231 B in the axial or z direction ( FIG. 2C ).
- voxels i.e., volume elements or three dimensional pixels
- the distances 260 A and 260 B between the centers of pixels 230 A and 230 B and the point 250 where the ray 220 intersects the horizontal line 240 passing through the center of pixels 230 A and 230 B define the interpolation coefficients in the x direction.
- the distances 261 A and 261 B between the centers of pixels 231 A and 231 B and the point 251 where the ray 220 intersects the horizontal line 241 passing through the center of pixels 231 A and 231 B define the interpolation coefficients in the axial direction.
- FIG. 3 shows an example of an oblique ray 320 in a rectangular image volume 310 for a full three-dimensional reconstruction. If one were to interpolate based on Joseph's Method as described above, both front (i.e. xy) surface 410 A and side (i.e. yz) surface 410 B projections of the oblique ray 320 (see FIGS. 4A and 4B ) would be necessary for each such oblique ray 320 , thus creating a front ray projection 420 A and a side ray projection 420 B.
- front ray projection 420 A of oblique ray 320 on front surface 410 A may coincide with the projection of a direct (i.e. two dimensional) ray on the same plane. Therefore, the interpolation coefficients in the x direction may be the same for front ray projection 420 A of oblique ray 320 and the direct two-dimensional ray. The pixel interpolation values for the direct rays thus could be reused on front ray projection 420 A.
- sheared volume 510 in each row from volume 310 of FIG. 3 , the vertical edges of the voxels may be skewed so that they are aligned with front ray projection 420 A on the xy surface.
- the two interpolations otherwise needed for oblique ray 320 may be reduced to a single interpolation of oblique ray 320 in sheared volume 510 .
- the interpolation coefficients may be the same for all the rays which differ only by their x coordinate. Therefore, only one interpolation coefficient can be used for all voxels of one axial row in the sheared volume. This coefficient may be different for each plane.
- FIG. 6 shows a three dimensional space 610 through which model ray 620 and model ray 630 pass.
- Model rays 620 and 630 have opposite polar angles (i.e. opposite angles in the y-z plane).
- rays 620 and 630 are projected onto the xy side surface 710 B (see FIG. 7B )
- model rays 620 and 630 are projected onto the yz front surface 710 A (see FIG. 7A ), it can be seen that they have opposite or mirrored yz front surface projections 720 A and 730 A.
- FIGS. 8A and 8B show front views of sheared volumes 810 A and 810 B for front projections 720 A and 730 A in accordance with the present invention. While sheared volumes 810 A and 810 B are different, each front projection 720 A and 730 A may coincide with a projection of a direct ray on the same plane. In practice, the same sheared volume may be used for both positive and negative polar angles, such that only one of the volumes 810 A and 810 B is actually necessary.
- the interpolation may reduce to a single interpolation of oblique model rays 620 and 630 in the sheared volume 810 A or 810 B, respectively. Since both model rays 620 and 630 have the same side projection 740 , both rays can be interpolated in the same single interpolation.
- ⁇ is the azimuthal and ⁇ is the polar angle.
- n′ integer part of y(x n )
- ⁇ n y(x n ) ⁇ n′.
- n′′ integer part of z(x n )
- a sheared volume is calculated using a 1D transaxial interpolation in the original volume. Because of the transaxial symmetry, the original and sheared volumes are stored with axial index first. An array of depth coordinates d is also computed, as such coordinates are used when computing interpolation factors for oblique segments. Projection rays are also stored with axial index first. The storage of the axial index as the first index is very important from a hardware point of view, as all operations are applied in axial direction first. Thus, having the axial index as the first index facilitates an efficient use of the memory cache and enables use of hardware parallelization. This results in fast computing. The projections for 2D segments are calculated at the same time as the sheared volumes. The projections for all oblique segments are then obtained by a 1D axial interpolation in the sheared volume.
- FIG. 9 shows an embodiment of a method 900 in accordance with the present invention.
- the first step 910 is to interpolate all the direct (i.e. planar) rays in the image volume.
- step 920 the front surface ray projections of the oblique rays may be matched to the direct rays.
- the voxel space may then be sheared at step 930 to align with the matched front ray projections.
- the oblique rays may be interpolated at step 940 in the sheared volumes.
- FIG. 10 is a system 1000 for using method 900 .
- System 1000 may be comprised of a medical imaging device 1010 , i.e. a PET scanner, a SPECT scanner or similar device capable of acquiring a medical image.
- Medical imaging device 1010 may be attached to a processor 1020 for receiving the data.
- Processor 1020 may have software running on it that executes a method of the present invention and outputs a fully three dimensional reconstruction of the object scanned.
- the method can be extended to a so-called LOR projection geometry when the transverse distance between rays is not a constant, as in a ring scanner. In such case, the method requires only a scanner with axial translation symmetry.
- the method also can be extended in the case of an unmatched back-projector. In such case, a different shear procedure would be used where each voxel receives contributions from two nearest projection rays in the transverse direction. This is important when the transverse voxel size is significantly smaller than the transverse projection size.
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Abstract
Description
y(x)=−cot(θ)x+y 0
or
x(y)=−y tan(θ)+x 0.
The above two equations are related in the interchange of x and y as independent and dependent variables.
where the terms T1 and TN represents the first and last pixel on the line and are treated separately, and λn is the fractional number defined by
n′=integer part of y(x n)
λn =y(x n)−n′.
ƒ(x n ,y(x n))≅(1−λn)P n,n′+λn P n,n′+1
and 2) implicitly in the sense that the summation above is the application of the trapezoidal rule to numerically estimate the one dimensional (x) integral.
P positive segment=value=w z*shearedvoxel(ρ,y,z)+(1−w z)* shearedvoxel(ρ,y,z+1)
While the same ray in the negative segment reuses the coefficients as:
P negative segment=(1−w z)*shearedvoxel(ρ,y,z)+w z*shearedvoxel(ρ,y,z+1)=shearedvoxel(ρ,y,z)+shearedvoxel(ρ,y,z+1)−value
where
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Cited By (4)
Publication number | Priority date | Publication date | Assignee | Title |
---|---|---|---|---|
US20110182491A1 (en) * | 2010-01-27 | 2011-07-28 | Levin Craig S | Shift-Varing Line Projection using graphics hardware |
US20120070050A1 (en) * | 2010-09-20 | 2012-03-22 | Siemens Medical Solutions Usa, Inc. | Time of Flight Scatter Distribution Estimation in Positron Emission Tomography |
WO2014016626A1 (en) | 2012-07-23 | 2014-01-30 | Mediso Orvosi Berendezés Fejlesztö És Szerviz Kft. | Method, computer readable medium and system for tomographic reconstruction |
US20190122399A1 (en) * | 2017-10-24 | 2019-04-25 | General Electric Company | Systems and methods for imaging with anisotropic voxels |
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CN103188998B (en) * | 2010-11-12 | 2015-03-04 | 株式会社日立医疗器械 | Medical image display device and medical image display method |
US8787644B2 (en) * | 2011-06-14 | 2014-07-22 | Kabushiki Kaisha Toshiba | Method and device for calculating voxels defining a tube-of-response using a central-ray-filling algorithm |
CN103218851B (en) * | 2013-04-03 | 2015-12-09 | 西安交通大学 | A kind of segment reconstruction method of three-dimensional line segment |
US9155514B2 (en) * | 2013-08-01 | 2015-10-13 | Siemens Medical Solutions Usa, Inc. | Reconstruction with partially known attenuation information in time of flight positron emission tomography |
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US20080180580A1 (en) * | 2007-01-25 | 2008-07-31 | Kadrmas Dan J | Rotate and slant projector for fast fully-3d iterative tomographic reconstruction |
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US20080180580A1 (en) * | 2007-01-25 | 2008-07-31 | Kadrmas Dan J | Rotate and slant projector for fast fully-3d iterative tomographic reconstruction |
Cited By (7)
Publication number | Priority date | Publication date | Assignee | Title |
---|---|---|---|---|
US20110182491A1 (en) * | 2010-01-27 | 2011-07-28 | Levin Craig S | Shift-Varing Line Projection using graphics hardware |
US9111381B2 (en) * | 2010-01-27 | 2015-08-18 | Koninklijke Philips N.V. | Shift-varying line projection using graphics hardware |
US20120070050A1 (en) * | 2010-09-20 | 2012-03-22 | Siemens Medical Solutions Usa, Inc. | Time of Flight Scatter Distribution Estimation in Positron Emission Tomography |
US8265365B2 (en) * | 2010-09-20 | 2012-09-11 | Siemens Medical Solutions Usa, Inc. | Time of flight scatter distribution estimation in positron emission tomography |
WO2014016626A1 (en) | 2012-07-23 | 2014-01-30 | Mediso Orvosi Berendezés Fejlesztö És Szerviz Kft. | Method, computer readable medium and system for tomographic reconstruction |
US20190122399A1 (en) * | 2017-10-24 | 2019-04-25 | General Electric Company | Systems and methods for imaging with anisotropic voxels |
US10482634B2 (en) * | 2017-10-24 | 2019-11-19 | General Electric Company | Systems and methods for imaging with anisotropic voxels |
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